34 research outputs found
Metrizable universal minimal flows of Polish groups have a comeagre orbit
We prove that, whenever is a Polish group with metrizable universal
minimal flow , there exists a comeagre orbit in . It then follows
that there exists an extremely amenable, closed, coprecompact of such
that
Topological properties of full groups
We study full groups of countable, measure-preserving equivalence relations. Our main results include that they are all homeomorphic to the separable Hilbert space and that every homomorphism from an ergodic full group to a separable group is continuous. We also find bounds for the minimal number of topological generators (elements generating a dense subgroup) of full groups allowing us to distinguish between full groups of equivalence relations generated by free, ergodic actions of the free groups F_m and F_n if m and n are sufficiently far apart. We also show that an ergodic equivalence relation is generated by an action of a finitely generated group if an only if its full group is topologically finitely generated
Modular actions and amenable representations
Consider a measure-preserving action Γ ↷ (X, μ) of a countable group Γ and a measurable cocycle α: X × Γ → Aut(Y) with countable image, where (X, μ) is a standard Lebesgue space and (Y, ν) is any probability space. We prove that if the Koopman representation associated to the action Γ ↷ X is non-amenable, then there does not exist a countable-to-one Borel homomorphism from the orbit equivalence relation of the skew product action Γ ↷^α X × Y to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets or, equivalently, an action on the boundary of a countably-splitting tree), generalizing previous results of Hjorth
and Kechris. As an application, for certain groups, we connect antimodularity to mixing conditions. We also show that any countable, non-amenable, residually finite group induces at least three mutually orbit inequivalent free,
measure-preserving, ergodic actions as well as two non-Borel bireducible ones